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EXAMPLE
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E.g.f.: A(x) = 1 + x + 7*x^2/2! + 133*x^3/3! + 4501*x^4/4! + 224497*x^5/5! +...
such that A(x) = exp(7*x*G(x)^6) / G(x)^6
where G(x) = 1 + x*G(x)^7 is the g.f. of A002296:
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
Note that
A'(x) = exp(7*x*G(x)^6) = 1 + 7*x + 133*x^2/2! + 4501*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 6*x^2/2 + 57*x^3/3 + 650*x^4/4 + 8184*x^5/5 + 109668*x^6/6 +...
and so A'(x)/A(x) = G(x)^6.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1, 7, 133, 4501, 224497, 14926387, 1245099709, ...];
n=2: [1, 2, 16, 308, 10360, 512624, 33845728, 2807075264, ...];
n=3: [1, 3, 27, 531, 17829, 876771, 57529143, 4745597787, ...];
n=4: [1, 4, 40, 808, 27184, 1331008, 86864512, 7129675840, ...];
n=5: [1, 5, 55, 1145, 38725, 1891205, 122869075, 10038831425, ...];
n=6: [1, 6, 72, 1548, 52776, 2575152, 166702752, 13564381824, ...];
n=7: [1, 7, 91, 2023, 69685, 3402679, 219682183, 17810832319, ...];
n=8: [1, 8, 112, 2576, 89824, 4395776, 283295488, 22897384832, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 27, 808, 38725, 2575152, 219682183, 22897384832, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 7^(n-5) * (n+1)^(n-6) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807) for n>=0.
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PROG
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(PARI) {a(n) = local(G=1); for(i=1, n, G=1+x*G^7 +x*O(x^n)); n!*polcoeff(exp(7*x*G^6)/G^6, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if(n==0|n==1, 1, sum(k=0, n, 7^k * n!/k! * binomial(7*n-k-7, n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))
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